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We present a discrete version of the two-dimensional nonlinear $O(3)$ sigmamodel examined by Belavin and Polyakov. We formulate it by means of Mercat'sdiscrete complex analysis and its elaboration by Bobenko and G\"unther. Wedefine a weighted discrete Dirichlet energy and area on a planar quad-graph andderive an inequality between them. We write $f$ for the complex functionobtained from the unit vector field of the model. The inequality is saturatedif and only if the $f$ is discrete (anti-)holomorphic. By using a weight $W$obtained from a kind of tiling of the sphere $S^2$, the weighted discrete area${\cal A}^{W}_{\diamondsuit}(f)$ admits a geometrical interpretation, namely,${\cal A}^{W}_{\diamondsuit}(f)=4 \pi N $ for a topological quantum number $N\in \pi_2(S^2)$. This ensures the topological stability of the solutiondescribed by the $f$, and we have the quantized energy$E^{W}_{\diamondsuit}(f)=|{\cal A}^{W}_{\diamondsuit}(f)|=4 \pi |N| $. Forquad-graphs with orthogonal diagonals, we show that the discrete(anti-)holomorphic function $f$ satisfies the Euler--Lagrange equation derivedfrom the weighted discrete Dirichlet energy. On some rhombic lattices, thediscrete power functions $z^{(N)}$ give the topological quantum number $N$.Moreover, the weighted discrete Dirichlet energy, area, and Euler--Lagrangeequation tend to their continuous forms as the lattice spacings tend to zero.

Nonlinear $O(3)$ sigma model in discrete complex analysis